It is a direct consequence of Furstenberg's correspondence principle [1]} that if the collection of sequences \(a_1,\ldots , a_\ell \)  is jointly ergodic, then every set of integers with positive upper density contains patterns of the form \(m,m+a_1(n),\ldots , m+a_\ell (n)\) , for some \(m,n\in {\mathbb {N}}\) .
In fact, a stonger property holds (which fails when say \(a_1(n)=n,a_2(n)=2n\) , see [2]}), namely, for every set of integers \(\Lambda \)  we have
\(\liminf _{N\rightarrow \infty }{\mathbb {E}}_{n\in [N]}\, \bar{d}(\Lambda \cap (\Lambda +a_1(n))\cap \cdots \cap (\Lambda +a_\ell (n)))\ge (\bar{d}(\Lambda ))^{\ell +1}.\) 
